Calculus Examples

$lim_{x \to \frac{x}{2}} \tan (x) \ln (\sin (x))$

Step 1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $\frac{x}{2}$ .

$lim_{x \to \frac{x}{2}} \tan (x) \cdot lim_{x \to \frac{x}{2}} \ln (\sin (x))$

Step 2

Move the limit inside the trig function because tangent is continuous.

$\tan (lim_{x \to \frac{x}{2}} x) \cdot lim_{x \to \frac{x}{2}} \ln (\sin (x))$

Step 3

Move the limit inside the logarithm.

$\tan (lim_{x \to \frac{x}{2}} x) \cdot \ln (lim_{x \to \frac{x}{2}} \sin (x))$

Step 4

Move the limit inside the trig function because sine is continuous.

$\tan (lim_{x \to \frac{x}{2}} x) \cdot \ln (\sin (lim_{x \to \frac{x}{2}} x))$

Step 5

Evaluate the limits by plugging in $\frac{x}{2}$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $\frac{x}{2}$ for $x$ .

$\tan (\frac{x}{2}) \cdot \ln (\sin (lim_{x \to \frac{x}{2}} x))$

Step 5.2

Evaluate the limit of $x$ by plugging in $\frac{x}{2}$ for $x$ .

$\tan (\frac{x}{2}) \cdot \ln (\sin (\frac{x}{2}))$

Step 6

Simplify the answer.

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Step 6.1

Rewrite $\tan (\frac{x}{2})$ in terms of sines and cosines.

$\frac{\sin (\frac{x}{2})}{\cos (\frac{x}{2})} \cdot \ln (\sin (\frac{x}{2}))$

Step 6.2

Combine $\frac{\sin (\frac{x}{2})}{\cos (\frac{x}{2})}$ and $\ln (\sin (\frac{x}{2}))$ .

$\frac{\sin (\frac{x}{2}) \ln (\sin (\frac{x}{2}))}{\cos (\frac{x}{2})}$

Step 6.3

Separate fractions.

$\frac{\ln (\sin (\frac{x}{2}))}{1} \cdot \frac{\sin (\frac{x}{2})}{\cos (\frac{x}{2})}$

Step 6.4

Convert from $\frac{\sin (\frac{x}{2})}{\cos (\frac{x}{2})}$ to $\tan (\frac{x}{2})$ .

$\frac{\ln (\sin (\frac{x}{2}))}{1} \tan (\frac{x}{2})$

Step 6.5

Divide $\ln (\sin (\frac{x}{2}))$ by $1$ .

$\ln (\sin (\frac{x}{2})) \tan (\frac{x}{2})$